Comparison Diagnosis Model Research Articles

Reliability analysis of godan graphs

The connectivity and diagnosability of a system or an interconnection network are two important measures. As a topology structure of interconnection networks, the godan graph EAn has many good properties. In this paper, we give various connectivity of EAn and two diagnosability of EAn under the comparison diagnosis model.

Hybrid Fault Diagnosis Capability Analysis of Highly Connected Graphs

Abstract Zhu et al. introduced the $h$-edge tolerable diagnosability to measure the fault diagnosis capability of a multiprocessor system with faulty links. This kind of diagnosability is a generalization of the concept of traditional diagnosability. A graph is called a maximally connected graph if its minimum degree equals its vertex connectivity. It is well-known that many irregular networks are maximally connected graphs and the $h$-edge tolerable diagnosabilities of these networks are unknown, which is our motivation for research. In this paper, we obtain the lower bound of the $h$-edge tolerable diagnosability of a class of $t$-connected graphs and establish the $h$-edge tolerable diagnosability of a class of maximally connected graphs under the PMC model and the MM$^*$ model, which extend some results in (Hakimi, S.L. and Amin, A.T. (1974) Characterization of connection assignment of diagnosable systems. IEEE Trans. Comput., 23, 86–88), (Chang, C.P., Lai, P.L., Tan, J.J.M. and Hsu, L.H. (2004) Diagnosability of t-connected networks and product networks under the comparison diagnosis model. IEEE Trans. Comput., 53, 1582–1590) and (Lian, G., Zhou, S., Hsieh, S.Y., Liu, J., Chen, G. and Wang, Y. (2019) Performance evaluation on hybrid fault diagnosability of regular networks. Theoret. Comput. Sci., 796, 147–153).

Diagnosability of the Cayley Graph Generated by Complete Graph with Missing Edges under the MM$^{\ast }$ Model

Abstract Diagnosability of a multiprocessor system is an important research topic. The system and an interconnection network have an underlying topology, which is usually presented by a graph. Under the Maeng and Malek's (MM) model, to diagnose the system, a node sends the same task to two of its neighbors, and then compares their responses. The MM$^{*}$ is a special case of the MM model and each node must test all pairs of its adjacent nodes. In 2009, Chiang and Tan (Using node diagnosability to determine $t$-diagnosability under the comparison diagnosis (cd) model. IEEE Trans. Comput., 58, 251–259) proposed a new viewpoint for fault diagnosis of the system, namely, the node diagnosability. As a new topology structure of interconnection networks, the nest graph $CK_{n}$ has many good properties. In this paper, we study the local diagnosability of $CK_{n}$ and show it has the strong local diagnosability property even if there exist $(\frac{n(n-1)}{2}-2)$ missing edges in it under the MM$^{*}$ model, and the result is optimal with respect to the number of missing edges.

Conditional diagnosability of the SPn graphs under the comparison diagnosis model

The conditional diagnosability of a system refers to the maximum cardinality of the conditional faulty sets can be identified by the system. The problem of conditional diagnosability is discussed widely. In this paper, we analyze the extra connectivities and the conditional diagnosability of a family of networks, called the SPn graphs. Applying the result, we determine the g-extra connectivity (1 ≤ g ≤ 3) and the conditional diagnosability of the star graphs and the pancake graphs under the comparison diagnosis model, respectively.

Conditional $(t,k)$ -Diagnosis in Regular and Irregular Graphs Under the Comparison Diagnosis Model

Assume that there are at most $t$ faulty vertices. A system is conditionally ( $t,k$ )- diagnosable if at least $k$ faulty vertices (or all faulty vertices if fewer than $k$ faulty vertices remain) can be identified in each iteration under the assumption that every vertex is adjacent to at least one fault-free vertex. Let $\kappa _c(G)$ be the conditional vertex connectivity of $G$ , which measures the vertex connectivity of $G$ according to the assumption that every vertex is adjacent to at least one fault-free vertex. Let $\Delta (G)$ be the maximum degrees of the given graph $G$ . When a graph $G$ satisfies the condition that for any pair of vertices with distance two has at least two common neighbors in $G$ , we show the following two results: 1) An $r$ -regular network $G$ containing $N$ vertices is conditionally $\left(\frac{N+\sqrt{\frac{4\kappa (G)N}{(r+1)(r-1)}}-2}{r+1},\kappa _c(G)\right)$ -diagnosable, where $r \geq 3$ and $N \geq \frac{(r+1)(25r-9)}{4\kappa (G)}$ . 2) An irregular network $G$ containing $N$ vertices is conditionally $(\frac{N}{\Delta (G)+1}-1,\kappa _c(G))$ -diagnosable. By applying the above results to multiprocessor systems, we can measure conditional $(t,k)$ -diagnosabilities for augmented cubes, folded hypercubes, balanced hypercubes, and exchanged hypercubes.

Improved Precise Fault Diagnosis Algorithm for Hypercube-Like Systems Based on the Comparison Diagnosis Model

Multiprocessor systems are being increasingly adopted and the system reliability is an important perspective for multiprocessor systems. The fault diagnosis has become crucial for achieving high reliability in multiprocessor systems. The precise fault diagnosis diagnoses all processors correctly. In the comparison-based model, it allows a processor to perform diagnosis by contrasting the responses from a pair of neighboring processors through sending the identical assignment. On the basis of comparison-based model, Sengupta and Dahbura (“On self-diagnosable multiprocessor systems: diagnosis by the comparison approach,” IEEE Transaction on Computers, vol. 41, no. 11, pp. 1386–1396, 1992) put forward the MM* model, any processor c diagnoses two processors c1 and c2 if c has direct communication links to them. Sengupta and Dahbura also designed an O(N5)-time precise fault diagnosis algorithm to diagnose faulty processors for general topologies by using the MM* model, where N is the cardinality of processor set in multiprocessor systems. Lately, Ye and Hsieh (“A scalable comparison-based diagnosis algorithm for hypercube-like net-works,” IEEE Transaction on Reliability, vol. 62, no. 4, pp. 789–799, 2013) devised an precise fault diagnosis algorithm to diagnose all faulty processors for hypercube-like networks by using the MM* model with O(N(log2N)2) time complexity. On the basis of Hamiltonian cycle properties, we improve the aforementioned results by presenting an O(N)-time precise fault diagnosis algorithm to diagnose all faulty processors for hypercube-like networks by using the MM* model.

Diagnosability and Diagnostic Algorithm for Pancake Graph under the Comparison Model

Fault diagnosis is important for the reliability of interconnection networks. This paper addresses the fault diagnosis of n-dimensional pancake graph Pn under the comparison diagnosis model. By the concept of local diagnosability, we first prove that the diagnosabitly of Pn is n − 1, and it has strong local diagnosability property even if there are n − 3 faulty edges. Furthermore, we present efficient algorithms to locate extended star and Hamiltonian path structures in Pn, respectively. According to the works of Li et al. and Lai, the extended star and Hamiltonian path structures can be used to identify all faulty vertices in linear time, provided the number of faulty vertices is no more than n − 1.

Conditional Diagnosability of <formula formulatype="inline"><tex Notation="TeX">$(n,k)$</tex></formula>-Star Networks Under the Comparison Diagnosis Model

The $(n,k)$ -star graph, denoted by $S_{n,k}$ , is an enhanced version of $n$ -dimensional star graphs $S_{n}$ , that has better scalability than $S_{n}$ , and possesses several good properties, compared with hypercubes. Diagnosis has been one of the most important issues for maintaining multiprocessor-system reliability. Conditional diagnosability, which is more general than classical diagnosability, measures the multiprocessor-system diagnosability under the assumption that all neighbors of any processor in the system cannot fail simultaneously. In this paper, we investigate the conditional diagnosability of $S_{n,k}$ for ( $n\geq 3$ and $k=1$ ) and ( $n\geq 4$ and $2\leq k\leq n$ ) under the comparison diagnosis model.

Connectivity Results of Complete Cubic Networks as Associated with Linearly Many Faults

We propose the complete cubic network structure to extend the existing class of hierarchical cubic networks, and establish a general connectivity result which states that the surviving graph of a complete cubic network, when a linear number of vertices are removed, consists of a large (connected) component and a number of smaller components which altogether contain a limited number of vertices.As applications, we characterize several fault-tolerance properties for the complete cubic network, including its restricted connectivity, i.e., the size of a minimum vertex cut such that the degree of every vertex in the surviving graph has a guaranteed lower bound; its cyclic vertex-connectivity, i.e., the size of a minimum vertex cut such that at least two components in the surviving graph contain a cycle; its component connectivity, i.e., the size of a minimum vertex cut whose removal leads to a certain number of components in its surviving graph; and its conditional diagnosability, i.e., the maximum number of faulty vertices that can be detected via a self-diagnostic process, in terms of the common Comparison Diagnosis model.

Conditional diagnosability of optical multi-mesh hypercube networks under the comparison diagnosis model

Due to integrated positive features of both hypercubes and tori, optical multi-mesh hypercube (OMMH) networks are regarded as a class of promising optical interconnection topologies. The notion of conditional diagnosability helps enhance the self-diagnosing capability of multicomputers. This paper determines the conditional diagnosabilities of OMMH networks under the Maeng–Malek comparison model.

Conditional (t,k)-Diagnosis in Graphs by Using the Comparison Diagnosis Model

(t, k)-Diagnosis, which is a generalization of sequential diagnosis, requires that at least k faulty processors be identified and repaired in each iteration when there are at most t faulty processors, where t ≥ k. Based on the assumption that each vertex is adjacent to at least one fault-free vertex, the conditional (t, k)-diagnosis of graphs was investigated by using the comparison diagnosis model. Lower bounds on the conditional (t, k)-diagnosability of graphs were derived, and applied to obtain the following results. 1) Symmetric d-dimensional grids are conditionally (N/2d+1 -1, 2d -1)-diagnosable when d ≥ 2 and N (the number of vertices) ≥ 4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> . 2) Symmetric d-dimensional tori are conditionally (1/5 (N + min{8/5 N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2/3</sup> , 2N-20/15} - 2), 6)-diagnosable when d = 2 and N ≥ 49 and ( N/2d+1 -1, 4d-2)-diagnosable when d ≥ 3 and N ≤ 4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> . 3) Cube-connected cycles are conditionally (N/4 - 1, 4)-diagnosable. 4) k-ary trees are conditionally (N/k+1 - 1)-diagnosable.

Local Diagnosis Algorithms for Multiprocessor Systems Under the Comparison Diagnosis Model

An efficient diagnosis is very important for a multiprocessor system. The ability to identify all the faulty devices in a multiprocessor system is known as diagnosability. In the comparison model, the diagnosis is performed by sending two identical signals from a processor to a pair of distinct neighbors, and then comparing their responses. Sengupta and Dahbura proposed a polynomial-time algorithm with time complexity O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> ) to diagnose a system with a total number N of processors under the comparison model. Recently, some concepts, such as the conditional diagnosability and the local diagnosability, are concerned with the measure which is able to better reflect fault patterns in real systems. In this paper, we propose a specific structure, the balanced wind-bell-tree, and give an algorithm to determine the fault status of each processor for conditional local diagnosis under the comparison model. According to our results, a specific t-connected network with the balanced wind-bell-tree structure is conditionally (2t-1)°-diagnosable, and the time complexity to diagnose all the faulty processors is O(N(logN) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) with our algorithm, where N is the total number of the processors in the network.

Diagnosability of optical multi-mesh hypercube networks under the comparison diagnosis model

Diagnosability of a multiprocessor system is one important study topic in the parallel processing area. As a family of promising optical interconnection topologies for massively parallel computers, the optical multi-mesh hypercube (OMMH) networks integrate positive features of both hypercube and mesh topologies and circumvent the lack of scalability of hypercubes and the large diameter of meshes. This paper studies an (l, m, n)-OMMH network and, proves that its diagnosability under the comparison diagnosis model is n+4 for l≥5, m≥5, n≥3.

Strong Diagnosability and Conditional Diagnosability of Multiprocessor Systems and Folded Hypercubes

Using the comparison diagnosis model, this study proposes some useful sufficient conditions for determining the strong diagnosability ts(G) and the conditional diagnosability tc(G) of a system G. Applying these results to an n-dimensional folded hypercube FQn shows that ts(FQn) = n + 1 for n ≥ 5 and tc(FQn) = 3n - 2 for n ≥ 5. Moreover, tc(FQ3) = 3 and tc(FQ4) = 7.

Component-Composition Graphs: (t,k)-Diagnosability and Its Application

(t,k)-Diagnosis, which is a generalization of sequential diagnosis, requires that at least k faulty processors be identified and replaced in each iteration provided there are at most t faulty processors, where t ≥ k. Let \kappa (G) and n(G) be, respectively, the node connectivity and the number of nodes in a graph G. In this paper, we compute the (t,k)-diagnosability for a class of component-composition graphs under the comparison diagnosis model. We show that the m-dimensional component-composition graph G (m ≥ 4) is (Ω (h), κ (G))-diagnosable, where h= {(2m-2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m-1</sup> )/2m<;sup>-1<;/sup>×lg(m-1)} if|V(G)|≥m! if 2<;sup>m-1<;/sup>≤|V(G|<;m! and κ(G) and |V(G)|denote the node connectivity and the number of nodes in G, respectively. Based on this result, the (t,k)-diagnosability of several multiprocessor systems can be computed efficiently.

The Conditional Diagnosability of k-Ary n-Cubes under the Comparison Diagnosis Model

Processor fault diagnosis plays an important role in measuring the reliability of multiprocessor systems and diagnosing many well-known interconnection networks. Conditional diagnosability is a novel measure of diagnosability that adds the additional condition that any faulty set cannot contain all of the neighbors of any vertex in a system. This study investigates some topological properties of k-ary n-cubes, where k ≥ 4 and n ≥ 4, and shows that the conditional diagnosability of k-ary n-cubes under the comparison diagnosis model is 6n - 5.

A general technique to establish the asymptotic conditional diagnosability of interconnection networks

We develop a general and demonstrably widely applicable technique for determining the asymptotic conditional diagnosability of interconnection networks prevalent within parallel computing under the comparison diagnosis model. We apply our technique to replicate (yet extend) existing results for hypercubes and k-ary n-cubes before going on to obtain new results as regards folded hypercubes, pancake graphs and augmented cubes. In particular, we show that the asymptotic conditional diagnosability of: folded hypercubes {FQn} is 3n−2, pancake graphs {Pn} is 3n−7, and augmented cubes {AQn} is 6n−17. We demonstrate how our technique is independent of structural properties of the interconnection network G in question and essentially only dependent upon the minimal size of the neighbourhood of a path of length 2 in G, the number of neighbours any two distinct vertices of G have in common, and the minimal degree of any vertex in G.

Strong Diagnosability and Conditional Diagnosability of Augmented Cubes Under the Comparison Diagnosis Model

The problem of fault diagnosis has been discussed widely, and the diagnosability of many well-known networks has been explored. Strong diagnosability, and conditional diagnosability are both novel measurements for evaluating reliability and fault tolerance of a system. In this paper, some useful sufficient conditions are proposed to determine strong diagnosability, and the conditional diagnosability of a system. We then apply them to show that an n-dimensional augmented cube AQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> is strongly (2n -1)-diagnosable for n ≥ 5, and the conditional diagnosability of AQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> is 6n - 17 for n ≥ 6. Our result demonstrates that the conditional diagnosability of AQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> is about three times larger than the classical diagnosability.

(t,k)-Diagnosis for Component-Composition Graphs under the MM* Model

(t, k)-Diagnosis, which is a generalization of sequential diagnosis, requires that at least t faulty processors be identified and replaced in each iteration provided there are at most t faulty processors, where t ≥ k. Let κ(G) and n(G) be, respectively, the node connectivity and the number of nodes in a graph G. In this paper, we compute the (t, k)-diagnosability for a class of component composition graphs under the comparison diagnosis model. We show that the m-dimensional component-composition graph G (m ≥ 4) is (Ω(h),κ(G))-diagnosable, where h= {2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m-1</sup> × (m-3) × lg(m-1) (m-1)/(m-1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> if 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m-2</sup> ≤ n(G)<; m! 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m-1</sup> × (m-3)/m-1 if n(G) ≥ m!. Based on this result, the (t, k)-diagnosability of several multiprocessor systems, including hypercubes, crossed cubes, twisted cubes, locally twisted cubes, multiply twisted cubes, generalized twisted cubes, recursive circulants, Mobius cubes, Mcubes, star graphs, bubble-sort graphs, pancake graphs, and burnt pancake graphs, can be computed efficiently.

The conditional diagnosability of crossed cubes under the comparison model

The growing size of the multiprocessor systems increases their vulnerability to component failures. It is crucial to locate and replace the faulty processors to maintain the system's high reliability. The fault diagnosis is the process of identifying faulty processors in a system through testing. The conditional diagnosis requires that for each processor v in a system, all the processors that are directly connected to v do not fail simultaneously. In this paper, we show that the conditional diagnosability of the crossed cubes CQ n under the comparison diagnosis model is 3n−5 when n≥7. Hence, the conditional diagnosability of CQ n is three times larger than its classical diagnosability.